Zuhair Abdul Ghafoor Al-Johar writes that:
"(1) Frege-Russell's Cardinals:
Cardinality(A) is the class of all sets equinumerous to A.
Those are incompatible with Z, since
they entail the existence of the set of all sets in Z, or in
NBG\MK they would be proper classes. However in
NF and related systems they are as general as the primitive
concept of Cardinality, but the problem with these theories is
that they are very complex, and difficult to understand, using
concepts of stratification of formulas which is not desirable,
even the finite axiomatization of NFU , though its axioms
do not use stratification, but yet most of its theorems
relies on it.
Those Cardinals were the first defined cardinals
in history of human kind."
Speaking strictly historical from the standpoint of the history of set
theory, it was of course Georg Cantor who first introduced cardinal
numbers and defined the concept of cardinality (as the "Machtigkeit" of
a set). Cantor's original treatment of the concept of cardinality and
his definition of cardinal numbers is to be found in his "Ein Beitrag
zur Mannigfaltigkeitslehre", Journal für die reine und angewandte
Mathematik 84 (1878), 242-258. A much more mature version is in his
"Beiträge zur Begründung der transfiniten Mengenlehre", Mathematische
Annalen XLVI (1895), 481-512.
Irving H. Anellis
Visiting Research Associate
Peirce Edition, Institute for American Thought
902 W. New York St.
Indiana University-Purdue University at Indianapolis
Indianapolis, IN 46202-5159
USA
URL: http://www.irvinganellis.info